|
%I #26 Oct 20 2023 15:54:18
%S 1,1,1,1,1,17,49,49,177,177,177,177,2225,2225,2225,18609,18609,84145,
%T 84145,84145,608433,1657009,1657009,1657009,1657009,1657009,1657009,
%U 1657009,135874737,404310193,941181105,2014922929,2014922929
%N Smallest positive integer solution x = a(n) of (3^4)*x - 2^n*y = 1 for n >= 0.
%C This is instance m=4 of the m-family of smallest positive solutions [x0(m,n), y0(m,n)] of 3^m*x - 2^n*y = 1, n >= 0, m >= 0, described in a comment on A239125.
%C The companion sequence is y(n) = y0(4, n) = A239131(n), which is periodic with period length phi(3^4) = 54, where phi(n) = A000010(n) (Euler's totient).
%C The G.f. can be found from that of the periodic sequence y(n).
%H Vincenzo Librandi, <a href="/A239130/b239130.txt">Table of n, a(n) for n = 0..1000</a>
%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1404.2710">On Collatz' Words, Sequences and Trees</a>, arXiv preprint arXiv:1404.2710 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Lang/lang6.html">J. Int. Seq. 17 (2014) # 14.11.7</a>
%F a(n) = (1 + 2^n*y0(4, n))/3^4, with y0(4, n) == ((3^4+1)/2)^(n + 3^3) (mod 3^4) = A239131(n), n >= 0.
%F a(n + 54) = 2^(54)*a(n) - (2^(54)-1)/3^4, n >= 0, from the y0(4, n) periodicity.
%e n=0: 81*1 - 1*80 = 1;
%e n=1: 81*1 - 2*40 = 1;
%e n=2: 81*1 - 4*20 = 1;
%e n=3: 81*1 - 8*10 = 1;
%e n=4: 81*1 - 16*5 = 1;
%e n=5: 81*17 - 32*5 =1; ...
%t Floor[Table[(2^n Mod[(41^(n + 27)), 81])/81 + 1, {n, 0, 40}]] (* _Vincenzo Librandi_, Mar 23 2014 *)
%o (Magma) [Floor(2^n*((41^(n+27) mod 81)/81))+1: n in [0..40]]; // _Vincenzo Librandi_, Mar 23 2014
%Y Cf. A000010, A007583 (m=1), A234038 (m=2), A239125 (m=3), A239131.
%K nonn,easy
%O 0,6
%A _Wolfdieter Lang_, Mar 22 2014
|
